Partially ordered rings and semi-algebraic geometry:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge
Cambridge Univ. Pr.
2007
|
| Edition: | First publ., reissued |
| Series: | London Mathematical Society: London Mathematical Society lecture note series.
37. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | 280 S. |
| ISBN: | 052122845X |
Staff View
MARC
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| 245 | 1 | 0 | |a Partially ordered rings and semi-algebraic geometry |
| 250 | |a First publ., reissued | ||
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Record in the Search Index
| _version_ | 1804141131053662208 |
|---|---|
| adam_text | Contents
Page
INTRODUCTION
...................................................... 1
CHAPTER I
-
PARTIALLY ORDERED RINGS
1.1.
Definitions
............................................... 32
1.2.
Existence of Orders
....................................... 33
1.3.
Extension and Contraction of Orders
....................... 34
1.4.
Simple refinements of orders
.............................. 36
1.5.
Remarks on the Categories (PORNN) and (PORCK)
............. 37
1.6.
Remarks on Integral Domains
............................... 39
1.7.
Some Examples
............................................. 40
CHAPTER II
-
1IOMOMORPIIISMS AND CONVEX IDEALS
2.1.
Convex Ideals and Quotient Rings
.......................... 45
2.2.
Convex Hulls
.............................................. 46
2.3.
Maximal Convex Ideals and Prime Convex Ideals
............. 49
2.4.
Relation between Convex Ideals in (A/P) and
(Λ/Ι/Ρ/Ι)
... 52
2.5.
Absolutely Convex Ideals
..................................
S2
2.6.
Semi-Noetherian Rings
..................................... 56
2.7.
Convex Ideals and Intersections of Orders
................. 62
2.8.
Some Examples
............................................. 66
CHAPTER
Ш
-
LOCALIZATION
3.1.
Partial Orders on Localized Rings
......................... 77
3.2.
Sufficiency of Positive Multiplicative Sets
............... 79
3.3.
Refinements of an Order Induced by Certain Localizations
.. 80
3.4.
Convex Ideals in
(Α, Ρ)
and
(AŢ,PT)
..................... 81
3.5.
Concave Multiplicative Sets
............................... 83
3.6.
The Shadow of
1 ............................:............. 84
3.7.
Localization at a Prime Convex Ideal
...................... 87
3.8.
Localization in (PORCK)
................................... 88
3.9.
Applications of Localization, I
-
Some Properties of Convex
Prime Ideals
........................................... 89
3.10.
Applications of Localization,
П
-Zero
Divisors
............ 91
3.11.
Applications
of Localization,
IH
-
Minimal Primes, Isolated
Sets of Primes, and Associated Invariants
.............
3.12.
Operators on the Set of Orders on a Ring
.................
96
CHAPTER IV
-
SOME CATEGORICAL NOTIONS
4.1.
Fibre Products
........................................... 101
4.2.
Fibre Sums
............................................... 102
4.3.
Direct and Inverse Limits
................................ 103
4.4.
Some Examples
............................................ 104
CHAPTER V
-
THE PRIME CONVF.X IDEAL SPECTRUM
5.1.
The Zariski Topology Defined
............................. 106
5.2.
Some Topological Properties
.............................. 107
5.3.
Irreducible Closed Sets in SpecCA.P)
.................... 107
5.4.
SpecCA/P) as a Punctor
.................................. 109
5.5. Disconnectedness of SpecCA/JS)
........................... 109
5.6.
The Structure Sheaf,
I
-А
First Approximation on Basic
Open Sets
............................................. 112
5.7.
The Structure Sheaf, II
-
The Sheaf Axioms for Basic Open
Sets
.................................................. 113
5.8.
The Structure Sheaf,
Ш
-
Definition
......................115
CHAPTER VI
-
POLYNOMIALS
6.1.
Polynomials as Functions
.................................118
6.2.
Adjoining Roots
..........................................120
6.3.
A Universal Bound on the Roots of Polynomials
............123
6.4.
A Going-Up Theorem for Semi-Integral Extensions
........ 125
CHAPTER
VII
-
ORDERED FIELDS
7.1.
Basic Results
............................................130
7.2.
Function Theoretic Properties of Polynomials
............. 132
7.3.
Sturm s Theorem
..........................................135
7.4.
Dedekind Cuts; Archimedean and Non-Archimedean Extensions.
137
7.5.
Orders on Simple Field Extensions
........................ 140
7.6.
Total Orders and Signed Places
........................... 144
7.7.
Existence of Signed Places
....-...........................148
CHAPTER
VIII -
AFFINE SEMI
-ALGEBRAIC SETS
8.1.
Introduction and Notation
................................
I62
8.2.
Some Properties of RHJ-Algebras
..........................
I68
8.3.
Real Curves
.............................................. 178
8.4.
Signed Places on Function Fields
......................... 184
8.5.
Characterization of Non-Negative Functions
............... 193
8.6.
Derived Orders
........................................... 196
8.7.
A Preliminary Inverse Function Theorem
................... 206
8.8.
Algebraic Simple Points, Dimension, Codimension and Rank
. 212
8.9.
Stratification of Semi-Algebraic Sets
.................... 218
8.10.
Krull Dimension
.......................................... 224
8.11.
Orders on Function Fields
................................ 232
8.12.
Discussion of Total Orders on R(x,y)
.................... 240
.13.
Brief
Discussion
of
Structure
Sheaves
.................... 247
I
-
The rational structure sheaf
......................... 248
II
-
The semi-algebraic structure sheaf
................... 252
IH
-
The smooth structure sheaf
........................... 262
APPENDIX. The Tarski-Seidenberg Theorem
................. 268
BIBLIOGRAPHY
............................................. 273
LIST OF NOTATION
......................................... 278
INDEX
.................................................... 279
|
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| genre | Semi-algebraische Geometrie gnd |
| genre_facet | Semi-algebraische Geometrie |
| id | DE-604.BV036076534 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T22:10:59Z |
| institution | BVB |
| isbn | 052122845X |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-018967699 |
| oclc_num | 634047347 |
| open_access_boolean | |
| owner | DE-739 |
| owner_facet | DE-739 |
| physical | 280 S. |
| publishDate | 2007 |
| publishDateSearch | 2007 |
| publishDateSort | 2007 |
| publisher | Cambridge Univ. Pr. |
| record_format | marc |
| series | London Mathematical Society: London Mathematical Society lecture note series. |
| series2 | London Mathematical Society: London Mathematical Society lecture note series. |
| spelling | Brumfiel, Gregory W. Verfasser aut Partially ordered rings and semi-algebraic geometry First publ., reissued Cambridge Cambridge Univ. Pr. 2007 280 S. txt rdacontent n rdamedia nc rdacarrier London Mathematical Society: London Mathematical Society lecture note series. 37. Geordneter Ring (DE-588)4156754-7 gnd rswk-swf Semialgebraischer Raum (DE-588)4116475-1 gnd rswk-swf Semi-algebraische Geometrie gnd rswk-swf Semi-algebraische Geometrie f DE-604 Geordneter Ring (DE-588)4156754-7 s Semialgebraischer Raum (DE-588)4116475-1 s London Mathematical Society: London Mathematical Society lecture note series. 37. (DE-604)BV000000130 37. Digitalisierung UB Passau application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018967699&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Brumfiel, Gregory W. Partially ordered rings and semi-algebraic geometry London Mathematical Society: London Mathematical Society lecture note series. Geordneter Ring (DE-588)4156754-7 gnd Semialgebraischer Raum (DE-588)4116475-1 gnd |
| subject_GND | (DE-588)4156754-7 (DE-588)4116475-1 |
| title | Partially ordered rings and semi-algebraic geometry |
| title_auth | Partially ordered rings and semi-algebraic geometry |
| title_exact_search | Partially ordered rings and semi-algebraic geometry |
| title_full | Partially ordered rings and semi-algebraic geometry |
| title_fullStr | Partially ordered rings and semi-algebraic geometry |
| title_full_unstemmed | Partially ordered rings and semi-algebraic geometry |
| title_short | Partially ordered rings and semi-algebraic geometry |
| title_sort | partially ordered rings and semi algebraic geometry |
| topic | Geordneter Ring (DE-588)4156754-7 gnd Semialgebraischer Raum (DE-588)4116475-1 gnd |
| topic_facet | Geordneter Ring Semialgebraischer Raum Semi-algebraische Geometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018967699&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000130 |
| work_keys_str_mv | AT brumfielgregoryw partiallyorderedringsandsemialgebraicgeometry |